SCALABLE OPTIMAL ONLINE AUCTIONS - web.stanford.edu

SCALABLE OPTIMAL ONLINE AUCTIONS

DOMINIC COEY, BRADLEY J. LARSEN, KANE SWEENEY, AND CAIO WAISMAN

ABSTRACT. This paper studies reserve prices computed to maximize the expected profit of

the seller based on historical observations of the top two bids from online auctions in an

asymmetric, correlated private values environment. This direct approach to computing re-

serve prices circumvents the need to fully recover distributions of bidder valuations. We

specify precise conditions under which this approach is valid and derive asymptotic prop-

erties of the estimators. We demonstrate in Monte Carlo simulations that directly estimating

reserve prices is faster and, outside of independent private values settings, more accurate

than fully estimating the distribution of valuations. We apply the approach to e-commerce

auction data for used smartphones from eBay, where we examine empirically the benefit of

the optimal reserve as well as the size of data set required in practice to achieve that benefit.

This simple approach to estimating reserves may be particularly useful for auction design

in Big Data settings, where traditional empirical auctions methods may be costly to imple-

ment, whereas the approach we discuss is immediately scalable.

Date: December 2, 2020.We thank the editor, the associate editor and two anonymous referees for their detailed comments and feed-back, and Isa Chaves, Han Hong, Dan Quint, Evan Storms, and Anthony Zhang for helpful comments. Thispaper was previously circulated under the title ”The Simple Empirics of Optimal Online Auctions.” Coey andSweeney were employees of eBay Research Labs while working on this project, and Larsen was a part-timecontractor when the project was started.Coey: Facebook, Core Data Science; [email protected]: Stanford University and NBER; [email protected]: Uber, Marketplace Optimization; [email protected]: Northwestern University; [email protected].

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2 COEY, LARSEN, SWEENEY, AND WAISMAN

1. INTRODUCTION

Auctions are a key selling mechanism in online markets. Not only are they employed

to sell physical goods through sites such as eBay and Tophatter, and many others, but

also to offer advertisement space online, such as through Google’s DoubleClick.1 A large

structural literature in economics has studied these auctions, but this literature offers little

practical advice for sellers wishing to determine the optimal reserve price. The traditional

approach to structurally analyzing auctions in the economics literature is to estimate the

full distribution of bidder valuations and only then compute counterfactual objects of in-

terest, such as the optimal reserve price or other instruments of auction design. While

off-the-shelf tools exist to estimate valuation distributions in many first-price auction set-

tings (e.g. Guerre et al. 2000), such tools do not immediately apply to online auction set-

tings, where the researcher/seller may not observe all bids or the number of bidders. Ex-

isting tools for estimating valuation distributions in online auctions are computationally

involved and have only been derived for limited settings, such as independent private

values (IPV) settings with symmetric bidders (e.g. Song 2004). In this paper, we adopt an

approach advocated in the computer science and statistical learning literature (e.g. Mohri

and Medina 2016): directly estimating optimal reserve prices without fully estimating un-

derlying distributions of bidder valuations. Such a direct approach is simple for sellers to

execute and scalable to large datasets and real-time computation without fully estimating

distributions of bidder valuations.

The primary information environment we consider is a private values setting with bid-

ders values being non-independent and potentially asymmetric. Our main focus is on on-

line auctions that follow a second-price-auction-like format but with sequential arrival of

bidders, such as those conducted on eBay. In such auctions, a bidder who arrives at the

auction after the standing bid has passed her valuation is not observed bidding. Hence,

1Given the large size of eBay alone, even with the decline in auction usage in recent years relative to postedprices or bargaining (Einav et al. 2018; Coey et al. 2020; Backus et al. 2020), auctions remain a popular mech-anism among many eBay users; as documented in Coey et al. (2020), as of 2014, eBay auctions annually soldover 80 million new-in-box items via auctions annually, yielding 1.6 billion dollars in annual revenue. Thesenumbers are even larger for used auctions. In some categories, such as event tickets, auctions remain the dom-inant selling mechanism (see Waisman 2020). In addition to eBay and Tophatter, other similar online auctionsites include eBid, ShopGoodwill, Webstore, Auction Zip, Catawiki, and Bid For Wine. Online auctions arealso used for some government surplus or government/police-apprehended items on sites such as GovdDeal,PropertyRoom, Municibid, and IRS auctions.

SCALABLE OPTIMAL ONLINE AUCTIONS 3

some bids, as well as the true number of would-be bidders, is unobserved to the econo-

metrician.2 This data limitation precludes the use of popular nonparametric approaches

for point identifying or partially identifying valuation distributions in ascending auctions,

which rely on inverting order statistic distributions (e.g., Haile and Tamer 2003; Athey and

Haile 2007; Aradillas-Lopez et al. 2013; Coey et al. 2017). Indeed, the full distribution of

bidder valuations is not identified in the setting we study, but the optimal reserve price

itself is identified. Specifically, in eBay-like auctions, a bid placed by the highest bidder is

recorded (unlike in traditional ascending/English auctions, where the auction ends with-

out the winner’s drop-out price being recorded). This opens up a direct approach to ob-

taining the information required for computing optimal reserve prices. Our starting point

is the observation that seller profit is a simple-to-compute function of the reserve price

and the two highest bids—and thus profit can be computed from only the top two bids

and without observing the number of bidders and without observing other losing bids.

The seller’s primary instrument of auction design in the real world is typically a sin-

gle reserve price.3 We demonstrate that, under mild conditions, the optimal single reserve

price can be computed by simply maximizing the seller expected profit function (as a func-

tion of the empirical distribution of historical observations of the two highest bids) with

respect to the reserve price. Throughout the paper, we treat the seller as synonymous with

the practitioner/researcher/econometrician.

We derive several properties of estimated reserve prices. We prove consistency and

derive the asymptotic distribution of the estimator of the optimal reserve price, which is

non-normal. Our interest in deriving these results is not just theoretical but also practical:

we wish to provide a clear notion of how many previous transactions the practitioner needs

to observe in order for it be the case that designing an auction based on estimated reserves

is a good idea, as well as guidance to perform statistical inference. In this spirit, building

on the work of Mohri and Medina (2016), we also derive an explicit lower bound, based

2A number of aspects of our analysis can also be applied to ad auctions, which have traditionally been runas second-price auctions (although a number of platforms have recently switched to first-price auctions) andwhere the true number of bidders may be unobserved due to the practice of marketing agencies bidding onbehalf of multiple bidders at once (Decarolis et al. 2020).3In theory, more complex auction design may be optimal—such as the Myerson (1981) auction for asymmetricindependent private values (IPV) settings—involving more than just a simple reserve price. And, in practice,other features of the auction beyond the reserve price can also impact revenue, such as the number of bidders(Bulow and Klemperer 1996; Coey et al. 2019) or the skill of the auctioneer (Lacetera et al. 2016). We do notfocus on those features here.


on only weak assumptions, for the number of auctions one would need to observe in order

to guarantee that the revenue based on the estimated reserve price approximates the true

optimal with a specified degree of accuracy.

In Monte Carlo simulations we demonstrate that the directly estimated optimal reserve

prices perform well relative to the approach of fully estimating the distribution of bidder

valuations (Song 2004). The latter approach is correct only if the true environment is one of

symmetric independent private values (IPV), whereas our approach does not rely on this

assumption. Furthermore, our approach is much faster and does not require specifying

any parametric approximation for the valuation distribution.

We then take a step beyond asymptotic and learning theory and examine empirically

the number of auctions one needs to observe in practice in order to achieve a level of rev-

enue close to the true optimal reserve price revenue. We analyze a sample of popular

smartphone products sold through eBay auctions. We find that implementing the optimal

reserve price in these settings would raise expected profit very little (less than 1%) com-

pared to an auction with a reserve price equal to the seller’s value. In contrast, in a setting

in which the seller plans to re-auction the item if the auction fails, the gains from optimal

reserve pricing are much larger (22–44%, depending on the product). In the data we find

that a historical dataset of less than 25 auctions is sufficient for estimating a reserve price

that will achieve a level of revenue that is within 1% of the true optimal-reserve revenue.

We also find that, from the perspective of a seller offering a limited quantity of inventory

(250 phones in our example), the seller would find it optimal to run fewer than 25 auctions

for the sole purpose of data collection before implementing a reserve price based on that

data. In online auctions for advertising or in e-commerce auctions for popular products,

such data requirements are not likely to be restrictive.

2. RELATED LITERATURE

Our work contributes to the literature studying empirical approaches for ascending-like

auctions. These methodologies generally rely on exploiting order statistics relationships.4

4A separate branch of the empirical auctions literature studies first-price auctions, such as Guerre et al. (2000)and a large work that builds on their approach, which exploits the relationship between equilibrium bidding


For example, in a symmetric, independent private values (IPV) button ascending auction,

the distribution of valuations is identified by inverting the distribution of the winning bid

(the second order statistic of valuations), as described in Athey and Haile (2002, 2007). Per-

forming such an inversion requires observing the number of would-be bidders. This object

is unobserved in eBay auctions. Song (2004) overcomes this challenge by pointing out that,

in a symmetric IPV online auction, two order statistics of bids are sufficient to identify the

underlying distribution of bidders, because the distribution of one order statistic condi-

tional on a lower order statistic does not depend on the number of bidders. This profound

insight led Hortacsu and Nielsen (2010) to argue that the Song (2004) result has long been

“the standard to beat in the empirical online auctions literature” due to its distinct ability

to identify the distribution of valuations when the number of bidders is unknown.

Like the Song (2004) approach, our method relies on observing two order statistics of

bids (in our case, the first and second highest). Our results are less useful than those

of Song (2004) in one sense, because we do not obtain identification of the distribution of

values, but our results are more useful in another sense, because we obtain identification of

one particular object of interest—the optimal reserve price—in environments that are more

general than the symmetric IPV auction of Song (2004). In particular, our results yield

identification of the optimal reserve price in online auctions with arbitrarily correlated

(and asymmetric) private values.

Song (2004) is the only approach of which we are aware that yields identification of

the full distribution of valuations when only two order statistics of bids are observed and

the number of bidders in unobserved. Other methodologies require stronger assumptions

on the information environment, such as assumptions on the form of any auction-level

unobserved heterogeneity (Freyberger and Larsen 2019; Luo and Xiao 2020), or stronger

assumptions on the data, such as partial knowledge of the distribution of the number of

bidders (Larsen and Zhang 2018; Hernandez et al. 2020; Larsen 2020), observability of

exogenously varying reserve prices (Freyberger and Larsen 2019), observability of more

than two bids or an instrument (Mbakop 2017), or observability of the number of bidders

(Luo and Xiao 2020). Our approach does not require any of these assumptions, but in

functions and valuations in first-price auctions. These approaches do not apply to ascending-like auctionenvironments.


exchange we only obtain identification of the optimal reserve price, not the full distribution

of valuations.

In the economics literature, the recent work of Aradillas-Lopez et al. (2013) provides

identification arguments for bounds on the optimal reserve price in symmetric correlated

private values environments using information on the number of bidders and only one bid

(the transaction price), and Coey et al. (2017) extends these arguments to asymmetric envi-

ronments. However, these approaches yield only partial—rather than point—identification,

and require the econometrician to observe the number of bidders. In this paper we instead

place a stronger data requirement on bids, requiring the top two bids be observed, and in

exchange we obtain point identification.

Our paper also relates to other empirical auction work in economics that, rather than

focusing on identification of the full distribution of valuations, provides direct inference

on identification of specific objects of interest, such as optimal reserve prices, seller profits,

bidder surplus, gains from optimal auction design, or losses from bidder mergers or collu-

sion. These papers include Li et al. (2003), Haile and Tamer (2003), Tang (2011), Aradillas-

Lopez et al. (2013), Chawla et al. (2014), Coey et al. (2017), and Coey et al. (2019). In this

sense, our work is also related to the broader movement in empirical economics work ad-

vocating for “sufficient statistics” for welfare analysis (Chetty 2009). This literature focuses

on obtaining robust implications for welfare, optimality, or other objects of interest from

simple empirical objects without requiring the type of detailed estimation typical of struc-

tural econometrics. In our setting, the sufficient statistics for the optimal reserve price are

the marginal distributions of the first and second-highest bids.5 Finally, our work is also

related to that of Ostrovsky and Schwarz (2016), where the authors experimentally vary re-

serve prices in position ad auctions to measure the improvement in profits from choosing

different reserve prices.

Our study contributes to the theoretical literature at the intersection of economics and

computer science. For example, a number of papers, surveyed by Roughgarden (2014),

5We also relate to other empirical work focusing specifically on online e-commerce auctions, such as Platt(2017). In our analysis we consider only private values settings; in theory work, Abraham et al. (2020) modelad auctions with common values. Our analysis in the Online Appendix, where we extend our approach toaddress the Myerson (1981) auction empirically, is related to Celis et al. (2014), which addresses non-regulardistributions and Myerson’s ironing in ad auctions, while we focus only on regular distributions.


examine approximately optimal auctions. One of the motivations for the focus on approxi-

mate optimality is that auction features, most notably reserve prices, are often estimated

from data and are therefore subject to sampling error. Thus, a lot of attention has been

devoted to deriving theoretical learning guarantees to approximate optimal auctions from

data as in Cole and Roughgarden (2014). Mohri and Medina (2016) propose algorithms

to compute reserves from historical data, incorporating auction-level covariates/features,

and these have been extended by Rudolph et al. (2016). A different approach consists of

tailoring algorithms to continuously update estimated reserve prices while data are col-

lected, an approach pioneered by Cesa-Bianchi et al. (2015) for symmetric IPV settings.

Other continuously updating methods include Austin et al. (2016) and Rhuggenaath et al.

(2019), both of which propose smooth approximations for the sellers profit function, deriv-

ing methods for handling high-dimensional auction-level observable features. We do not

consider controlling for auction-level observable features or dynamic updating of reserve

prices here. Kanoria and Nazerzadeh (2020) consider a model where bidders account for

the fact that their bids may be used to compute future reserve prices; we do not explicitly

model this possibility.6

The most closely related study to ours is that of Mohri and Medina (2016). Our approx-

imation of profits and the corresponding optimal reserve prices are equivalent to theirs, as

both papers estimate reserve prices using historical observations of the first- and second-

highest bids. Both studies propose searching for the optimal reserve price using a grid

search, and Mohri and Medina (2016) provide specific guidelines on how to perform this

search quickly. In one important dimension, the work of Mohri and Medina (2016) is more

general than ours, as the authors discuss controlling for auction-level observable covari-

ates in estimating the optimal reserve price, and we do not consider auction-level covari-

ates here; rather, we limit to commodity-like products and then obtain reserve prices that

average over any remaining auction-level heterogeneity, observable or unobservable to the

econometrician. Our approach is valid even in the presence of auction-level heterogeneity

(observable or unobservable), but our approach averages over this heterogeneity rather

than controlling for observable heterogeneity as in Mohri and Medina (2016).

6More broadly, our work also relates to the growing literature considering sophisticated pricing strategies andalgorithms available to sellers with access to Big Data (e.g. Bounie et al. 2020; Kehoe et al. 2020; Ali et al. 2019).


Our contribution relative to Mohri and Medina (2016) is threefold. First, we offer a

treatment of asymptotics and inference for the estimated reserve price, which Mohri and

Medina (2016) do not. Second, we describe underlying conditions on the auction envi-

ronment that are sufficient for the approach to be valid: a sequential-arrival second-price

auction with private values. In contrast, Mohri and Medina (2016) provide their results in

terms of bids and remain agnostic about the model generating those bids. In being explicit

about these conditions, we are also able to offer examples of specific settings in which the

approach is not valid, such as common values environments, dynamic auctions (i.e. set-

tings in which bidders can participate in another auction if they fail to win the current

auction), and certain types of endogenous entry. These are discussed in Section 3.6. Third,

we offer a comparison to the alternative of fully estimating the distribution of valuations.

Algorithmic-based studies in computer science and empirical auction studies in eco-

nomics tend to be quite distinct, and the two strands of literature do not often speak to one

another. A contribution of our work relative to each of the above computer science studies

is to help bridge this gap by bringing an econometrics and economic theory point of view

to the computer science algorithmic literature.

3. MODEL AND EMPIRICAL APPROACH

3.1. Model Overview. We consider bidders in an eBay-like auction, which we will refer to

as a sequential-arrival second-price auction. Each auction has a number of bidders N that will

arrive to the auction. N is a random variable potentially varying from auction to auction.

We do not model bidders’ choice of which of multiple auctions to participate in; rather, we

treat bidders as being assigned to one and only one auction, and we assume that after the

auction all N bidders assigned to that auction (the winner and the losers) exit the market.

The seller of the auction sets a reserve price, r ě 0.7 In a given auction, bidder i, where

i P t1, ..., Nu, has a valuation Vi ě 0 for the item. Let F denote the joint distribution of

bidder valuations from which N bids are drawn. Bidders arrive sequentially to the auction

and have a single opportunity to bid.8 Bidders see the current second-highest bid or, if no

bids have been placed exceeding r, bidders see r. At some set time, the auction ends. As

7eBay uses the term “start price” to refer to a public reserve price and the term “reserve price” to refer to asecret reserve price. We will simply use the term “reserve price” except when explicitly discussing a featureunique to secret reserve prices, in which case we will use the term “secret reserve price”.8This single-opportunity-to-bid assumption can be relaxed following the intuition in the random-bidding-opportunities model of Song (2004).


long as at least one bidder bids above the reserve price, the highest bidder wins and pays

the maximum of the second-highest bid and the reserve price.

We assume bidders’ valuations are private, and we allow for bidders’ valuations within

a given auction to be arbitrarily correlated and to be potentially drawn from different mar-

ginal distributions.9 Thus, the environment we consider is one of asymmetric correlated

private values. In this private values environment it is a weakly dominant strategy for

each bidder to bid her value when arriving at the auction if the current price is below her

value, and we assume all bidders play this weakly dominant strategy. This abstracts away

from any bid-sniping. We also abstract away from any cost of entering the auction and

placing a bid, and from minimum bid increments.

Each bidder who arrives at the auction before the current price exceeds her valuations

places a bid, and that bid is recorded by the platform. Each bidder who arrives after the

bidding passes her valuation places no bid, and the platform has no information on her

valuation. Under our modeling assumptions, in any auction in which at least two bidders

have valuations weakly higher than r, the top two highest valuations among the N bidders

will always be recorded, but lower order statistics of valuations may not be. For example,

the third-highest-value bidder will not be observed bidding in an auction in which the two

highest-valuation bidders arrive and bid their valuations before the third-highest-valuation

bidder arrives. Consider another example: r “ 0, and 10 bidders arrive at the auction. The

fourth-highest bidder arrives first, followed by the first-highest, followed by the second-

highest, followed by all other bidders in any arbitrary order. Only three bids are recorded

because all other bidders arrive after the bidding has passed their valuations. However,

among those three recorded bids, only the top two represent corresponding order statistics

of valuations: the third-highest bid in this example is equal to the fourth-highest valuation.

Our approach relies only on observing the top two bids, not other losing bids and not the

underlying number of bidders N. We also remain agnostic as to the precise arrival order

of the bidders. Under our above assumptions, the top two valuations will be correctly

observed regardless of the other details of the arrival process of bidders.

9In the Online Appendix we consider the asymmetric IPV environment of Myerson (1981). In this environ-ment, if, in addition to the two highest bids, the econometrician observes the identity of the highest bidder,then bidder-specific marginal revenue curves (as defined by Bulow and Roberts 1989) are identified and es-timable. These objects are then sufficient to implement the Myerson (1981) optimal auction. We use simulateddata to illustrate this approach and quantify the revenue gain from optimal auction design.


We denote the highest and second-highest values by Vp1q and Vp2q. In any auction in

which only one bidder arrives, we consider Vp2q to be equal to 0, and in any auction in

which no bidders arrive, we consider Vp1q “ Vp2q “ 0. Thus, Vp1q and Vp2q refer to the

top two order statistics of valuations within a given auction, whereas the set tViuNi“1 is the

unordered set of bidders’ valuations.

The seller’s profit from setting a reserve price r in such an auction is given by

πpVp1q, Vp2q, v0, rq “ r1pVp2q ă r ď Vp1qq `Vp2q1pr ď Vp2qq ` v01pr ą Vp1qq, (1)

where v0 indicates the seller’s value from keeping the good. Expected profits as a function

of the reserve are given by:

pprq ” E”

πpVp1q, Vp2q, v0, rqı

, (2)

where we suppress dependence of pp¨q on v0 for notational simplicity. The expectation is

taken with respect to the distribution of Vp1q and Vp2q. We use the term optimal reserve price

to refer to a reserve price maximizing pprq, and denote such a reserve by r˚.

3.2. Assumptions and Main Result. As highlighted above, in our environment, it is a

weakly dominant strategy for bidders to bid their valuations if their valuations are above

the current auction price. We explicitly assume that this is the bidding strategy that bidders

follow. We summarize this and all of our key model assumptions below:

Assumption 1.

(i) Bidders have private valuations.

(ii) Bidders play the weakly dominant strategy of bidding their valuations if their valuations are

above the current auction price.

(iii) Bidders are assigned to one auction and have one opportunity to bid in that auction.

(iv) Bidders face no cost of entry or bidding in the auction.

(v) PrpN “ n|rq “ PrpN “ nq for all r and all n, and F , the joint distribution of valuations, does

not vary with r.

Conditions (i)–(iv) are discussed above. Condition (v) ensures that if the auction de-

signer were to change the reserve price (from r “ 0 to r “ r˚, for example) this would

not change the unobserved number of bidders matched to the auction or the distribution


from which they draw their values. Under these assumptions, the following identification

result is immediate:

Remark 1. Under Assumption 1, the optimal reserve price, r˚, is identified from the marginal

distributions of the first and second-highest bids from auctions with r “ 0.

We can instead state a weaker (and less explicit) assumption under which Remark 1

holds. First, we define the concept of a would-be bid, which is the bid a bidder places in

an auction or, if she is unable to place a bid because of the current price or reserve price

exceeding her valuation, it is the bid she would have placed if not prevented from doing so.

Assumption 2. The marginal distributions of the first and second-highest would-be bids do not

vary with r.

Under this assumption, Remark 1 will also hold. This assumption means that the data

generating process for the historical auctions observed by the econometrician prior to im-

plementing the reserve price is the same as the data generating process for the would-be

bids after the reserve price is implemented. What we need for the estimated reserve price

to be valid is that the distributions of bids (more specifically, the top two bids) does not

change on the intensive margin when the reserve price is implemented, only on the ex-

tensive margin. In other words, the reserve price can determine whether a bidder places

their bid, but not what that bid is.10 Settings where our approach is not valid (discussed in

Section 3.6) are violations of this assumption.

We assume that the econometrician has access to a random sample of j “ 1, ..., J auctions

with zero reserve prices.11 For each auction j, we assume that the top two highest bids,

Vp1qj and Vp2qj , are observed. With these data, we construct the sample analog of pp¨q and

its maximizer, which is our estimator of r˚:

pprq ”1J

Jÿ

j“1

”

πpVp1qj , Vp2qj , v0, rqı

. (3)

rJ ” arg maxr

pprq. (4)

10This same assumption, although not explicitly stated, also underlies the optimal reserve price exercisesin other empirical auction work, such as Haile and Tamer (2003) and Aradillas-Lopez et al. (2013), and thecomputer science literature cited in Section 2.11Hence, the data are assumed to be independently and identically distributed across auctions.


We will also use the plug-in estimator pprJq as our estimator of ppr˚q.

The reserve price estimated by this maximizer is the correct optimal reserve price un-

der the assumptions of our model. Mohri and Medina (2016) advocate this same direct

approach: maximizing expected seller profit directly using historical observations of the

first- and second-highest bids rather than fully estimating the underlying distribution of

seller valuations.

3.3. Unobserved Auction-Level Heterogeneity. We now comment briefly on auction-level

unobservable heterogeneity—features of bidder valuations that are observable to all bid-

ders (and potentially the seller) and that are not observable to the econometrician. Our

proposed approach allows for the presence of such unobservable heterogeneity, as it sim-

ply introduces correlation between Vp1q and Vp2q. If such unobservables are present, the es-

timated reserve price from our approach simply averages over these unobservables, yield-

ing the optimal unconditional reserve price. What our approach does not allow for is the

estimation of the full distribution of bidder valuations or the full distribution of unobserved

heterogeneity. Other approaches do exist in the literature for estimating the full distri-

bution of bidder valuations and the full distribution of unobserved heterogeneity (such

as Freyberger and Larsen 2019), typically relying on convolution theorem arguments to

disentangle these distributions. These approaches allow the researcher to compute the op-

timal reserve price conditional on a realization of the unobserved heterogeneity component.

However, these approaches rely on stronger assumptions on the information environment

or on the data than our approach. For example, Freyberger and Larsen (2019) require data

on reserve prices and require assuming a scalar unobserved heterogeneity component in

bidder values that is additively or multiplicatively separable from the private-value com-

ponent. Our approach does not require these assumptions.

The way our approach handles unobserved auction-level heterogeneity by computing

an optimal reserve price that averages over realizations of unobserved heterogeneity is

similar to the way unobserved heterogeneity is handled in the bounds approaches of

Aradillas-Lopez et al. (2013) and Coey et al. (2017), which, like our paper, focus on cor-

related private values settings.12 The important point we wish to emphasize is that the

12The bounds in both of those previous papers apply in the case where the number of bidders and the second-highest bid are observed (but the first-highest bid is not), obtaining partial identification in those cases. Ourpaper, on the other hand, obtains point identified reserve prices for the case where the number of bidders is


presence of unobserved heterogeneity does not in any way invalidate our approach; it

simply means that the reserve prices delivered by our approach represent reserve prices

averaged over any unobservable heterogeneity. Like the approaches of Aradillas-Lopez

et al. (2013) and Coey et al. (2017), it is possible that the reserve prices from our approach

may underperform relative to a case in which the seller observes the realization of auction-

level unobserved heterogeneity and conditions on this realization in setting the reserve

price.

3.4. Public vs. Secret Reserve Prices. While Remark 1 refers to historical auctions with

no reserve price, our method is also valid using historical auctions with a positive secret

reserve price, because a secret reserve price would still allow the econometrician to observe

realizations of the first- and second-highest bids in each auction. In fact, there is a strong

argument for also using a secret reserve price to implement the optimal reserve price once

it has been estimated, as this would allow the econometrician to continue observing more

realizations of the first- and second-highest bids, further improving the accuracy of the

estimated reserve price.13

3.5. The Seller’s Valuation, v0. We do not propose any estimate of the parameter v0,

the seller’s valuation. We see this as a parameter of the problem known to the practi-

tioner/seller, and to be specified by her when estimating the optimal reserve price. For

example, a seller expecting to obtain $100 for an item if the current auction fails to close

should compute the optimal reserve price with v0 set to $100.14 A reasonable value for v0

unobservable but both the first- and second-highest bids are observable. In either case, unobserved auction-level heterogeneity only introduces correlation in private values and does not invalidate the methods.13An alternative but complementary argument underlies the seller’s motivation for using secret reserve pricesin Andreyanov and Caoui (2020): If bidders are better informed about auction-level heterogeneity than theseller, a seller may wish to observe a given auction first and then choose to accept or reject the auction priceonly after the bidding ends, effectively implementing a secret reserve price. In the environment we study, aseller could potentially benefit from using a secret reserve price to learn about bids for future auctions, ratherthan profiting from learning in the current auction as in Andreyanov and Caoui (2020).14Importantly, it is unnecessary to make any assumptions regarding what the seller’s valuation is in the histor-ical auction data; the pieces of information the seller/econometrician needs from these historical auctions arethe marginal distribution of the first- and second-highest bids; the parameter v0 only matters for computingthe optimal reserve price going forward, given these historical auction bids. Similarly, it is also unnecessary tomake any assumption regarding correlation (or lack thereof) between v0 and bidder valuations; indeed, for agiven seller computing the optimal reserve price, v0 is a scalar constant. Furthermore, the correlation structurebetween seller valuations and bidder valuations in the historical auctions has no effect on the bids because theprivate values environment ensures that each bidder plays the weakly dominant strategy of bidding her val-uation whenever the current auction price does not exceed her valuation, regardless of the seller’s valuationor the seller’s choice of the reserve price.


might be the seller’s estimate of the price at which the good might sell if the auction were

to fail (such as the expected price from recent posted-price listings in the eBay context,

which the seller can search on the eBay website). An alternative would be for the seller to

consider re-auctioning the item if it fails to sell. The following recursive formula specifies

the seller’s dynamic payoff πprq in this setting:

πprq “ Ermaxtr, Vp2qus ´ r PrpVp1q ă rq ` β PrpVp1q ă rqπprq

where β is the seller’s discount factor. Rearranging the above expression yields

πprq “1

1´ β PrpVp1q ă rq

´

Ermaxtr, Vp2qus ´ r PrpVp1q ă rq¯

(5)

Maximizing this expression yields an estimate of the optimal reserve price in this repeated

auction setting.15 This can be implemented empirically similarly to the main case of (3).

3.6. Limitations. There are several settings (ruled out by Assumption 1 or Assumption 2)

in which our identification and estimation approach would not be valid because the dis-

tribution of bids in auctions with no reserve price (which Remark 1 relies on) would not

be representative of the distribution of bids in auctions with a positive reserve price. First,

the reserve price estimated using historical bid data from zero-reserve auctions will not

necessarily be optimal in an interdependent/common values environment. In such an en-

vironment, the reserve price itself affects equilibrium bidding (as described, for example,

in Milgrom and Weber 1982; Cai et al. 2007; Quint 2017), and can provide signaling value

for bidders. Assumption 1(i) rules out this environment.

Second, our estimated reserve prices would not necessarily be valid if bidders can choose

which (of several) auctions to participate in, or can pass on a particular auction and await

a future auction. In such a multiple, simultaneous (or dynamic) auction environment, a

bidder’s choice of which auction to participate in would depend on the reserve price in

each auction, and hence the distribution of the two highest bids in an auction would also

depend on r. This would violate Assumption 2, and a seller using historical bid data from

15Note that we implicitly assume here a stationary dynamic environment, ruling out the idea that the gooditself is perishable (Waisman 2020).


auctions with r “ 0 using our approach would not infer the correct optimal reserve price.16

This type of environment is ruled out by Assumption 1(iii).17

Third, our model does not allow for certain types of endogenous entry. We explicitly

rule out entry costs in Assumption 1(iv) and we impose that the distribution of the number

of bidders does not depend on N in Assumption 1(v). Models of auctions with endoge-

nous entry typically consider a two-stage decision-making process: first, bidders decide

whether to participate in the auction, and then, conditional on participating, what bid to

place. Whether one assumes that valuations are known prior to the entry decision (Samuel-

son 1985) or after it (Levin and Smith 1994), the decision to participate is often given by a

threshold strategy: a bidder participates if and only if his expected payoff from the auc-

tion exceeds his participation (or entry) cost. In both cases, this expected payoff directly

depends on r, which therefore directly determines the expected number of bidders that

participate in the auction. Our approach can only allow for endogenous entry (as in the

aforementioned models) if bidders’ entry decisions are made before bidders observe r.

4. ASYMPTOTIC PROPERTIES OF ESTIMATED RESERVES AND REVENUE

In this section we discuss the properties of the estimators we propose for the optimal

profit and optimal reserve price, defined in equations (3) and (4). For ease of notation,

throughout this section we set v0 “ 0, but the results we now present do not require this.

All proofs are found in the Appendix.

We first present asymptotic results for the estimators rJ and pprJq. We maintain the

following technical assumptions:

Assumption 3.

(i) The joint distribution of Vp2q and Vp1q admits an absolutely continuous, Lebesgue-measurable

16Under such circumstances, determining the optimal reserve price requires re-solving the bidders’ optimiza-tion problem for a new equilibrium as in Balseiro et al. (2015) and Choi and Mela (2019), for example. In turn,determining an equilibrium solution for the bidders’ optimization problem in these dynamic environments re-quires stronger assumptions than Assumption 1. An even more general and complex model is that of Kanoriaand Nazerzadeh (2020), who consider forward-looking bidders that not only anticipate future reserve prices,but also internalize that their current bids will be used by the seller to estimate and implement future reserveprices.17A recent empirical and theoretical literature does allow for multiple, dynamic auctions and long-lived bid-ders, such as Zeithammer (2006), Hendricks and Sorensen (2018), Backus and Lewis (2020), Bodoh-Creed et al.(2020), and Coey et al. (2020).


density; (ii) 0 ď Vp2q ď Vp1q ď ω ă 8; (iii) The function pprq is uniquely maximized at r˚; (iv)

The function πp¨, rq ” πp¨, rq ´ πp¨, r˚q has Erπp¨, rqs twice differentiable.

These assumptions are standard technical conditions in the auction methodology lit-

erature and the econometrics literature more broadly. Conditions (i) and (ii) guarantee

continuity of pprq. Condition (iii) indirectly imposes restrictions on the underlying distri-

butions of valuations and bidder arrival process. In some information environments, it is

easy to specify sufficient conditions for (iii) to be satisfied. For example, in a setting with

symmetric independent private values, a monotone hazard rate for bidder valuations is a

sufficient condition for pprq to be uniquely maximized at r˚. Deriving a general sufficient

condition for (iii) to be satisfied is beyond the scope of this paper, and we therefore state

it directly as an econometric assumption.18 Condition (iv) (twice differentiability) will be

satisfied provided the joint density of Vp2q and Vp1q is sufficiently smooth.

Under these assumptions, we derive a number of results. First, we demonstrate consis-

tency of both rJ and pprJq:

Theorem 1.

If Assumptions 1 and 3 are satisfied, then pprJqpÝÑ ppr˚q and rJ

pÝÑ r˚.

The proof, given in the Appendix, consists of showing that, under Assumption 3,

suprPR |pprq ´ pprq|pÝÑ 0, where R ” r0, ωs. Hence, all the requirements from Theorem

2.1 of Newey and McFadden (1994) are satisfied, which yields Theorem 1.

Having established consistency, we now derive the asymptotic distribution of the esti-

mator rJ , which is not standard. The estimator belongs to a class of estimators that con-

verge at a cube-root rate, of which an example is the maximum score estimator proposed

by Manski (1975, 1985). To demonstrate this and derive the asymptotic distribution, we

show that the conditions in the main theorem of Kim and Pollard (1990) are satisfied.19

Theorem 2.

If Assumptions 1 and 3 are satisfied, and if r˚ is an interior point, then the process J2{3 1JřJ

j“1 πp¨, r˚`

18See van den Berg (2007) for a more extensive study that addresses sufficient conditions for a unique optimumin monopolistic selling problems.19This same result regarding the rate of convergence was obtained by Segal (2003) in the context of optimalmechanisms with unknown demand and by Prasad (2008) in the context of posted prices. While Segal (2003)only derived the rate of convergence, Prasad (2008) obtained the estimator’s asymptotic distribution in thesame way we do: by verifying that the conditions in Kim and Pollard (1990) are satisfied.


αJ´1{3q converges in distribution to a Gaussian process, and J1{3`

rJ ´ r˚˘

converges in distribu-

tion to the random maximizer of this process.

For additional technical details and discussion, we refer the interested reader to the Ap-

pendix. The important implication we highlight here is the cube-root rate of convergence

of rJ , discussed in more detail in the Appendix. This is slower than the square-root rate

typical to many econometric settings, and suggests a stronger data requirement to obtain

a precise estimate of r˚. We examine the practical relevance of these theoretical results in

our application in Section 6.

While accurately estimating the optimal reserve price itself may in theory require a large

dataset, determining how much the auction designer could gain from optimally choosing the

reserve price does not. In fact, pprJq converges at a square-root rate to a normal distribution,

which we present in the following theorem:

Theorem 3.

If the conditions from the previous theorems are satisfied, it then follows that?

J“

pprJq ´ ppr˚q‰ dÝÑ N p0, Varrπp¨, r˚qsq.

We now discuss inference. Simulating the asymptotic distribution of rJ is impractical

as the second derivative of the expected profit function depends upon the distributions

of the two order statistics used to estimate r˚. Resampling methods are a useful alter-

native to simulation. Abrevaya and Huang (2005) showed that the most straightforward

resampling method for inference, nonparametric bootstrap, is not valid for this cube-root

class of estimators. Alternative resampling methods that may be used in this case include

subsampling (Delgado et al., 2001), m out of n bootstrap (Lee and Pun, 2006), numerical

bootstrap (Hong and Li, 2020), and rescaled bootstrap (Cattaneo et al., 2020). Another

possibility is to replace the indicator in the objective function with a smoothed estimator,

which, along with further assumptions, might restore asymptotic normality and achieve

faster rates of convergence, akin to the smoothed maximum score estimator introduced by

Horowitz (1992). We leave this possibility as an avenue for future research.

However, the nonparametric bootstrap can be used for inference on the object pprJq. Let

pbprq be the objective function defined above calculated from a bootstrap sample of size J

drawn with replacement from the original sample, and let rbJ be the estimator calculated


from this bootstrap sample. Results in Abrevaya and Huang (2005) yield pbprbJq ´ pbpr˚q “

OPpJ´2{3q and rbJ ´ r˚ “ OPpJ´1{3q, which imply that

a

J”

pbprbJq ´ pprJq

ı

“a

J”

pbpr˚q ´ ppr˚qı

` oPp1q,

which, in turn, has the same limiting distribution as?

J“

pprJq ´ ppr˚q‰

conditional on the

data due to a standard result from bootstrap theory.

We now provide a theoretical argument establishing a probabilistic upper bound on the

quantity ppr˚q´ ppprJq, which shrinks to zero at the rate plog J{Jq´1{2. We seek an expression

that, under very minimal assumptions on the distribution of valuations, can aid the prac-

titioner in making an informed decision regarding the data collection process: how many

absolute (no reserve) auctions to run to obtain the estimate rJ .20 This argument builds on

arguments from Mohri and Medina (2016).21 The bound relies on virtually no assumptions

on the distribution of valuations (other than that they have a finite upper bound), and

as such it is very conservative, as is typical in the algorithmic game theory literature in

computer science. It yields a worst-case-scenario bound on revenue performance; as such,

the result will be most useful as a guide to the conservative practitioner. A second use of

the bound is that it allows us to provide a graphical illustration of how the accuracy of

estimated reserves can improve with the size of the dataset.

Given a sample of J auctions, we refer to the seller’s reserve price as being the estimated

reserve price if the seller chooses the reserve that maximizes profit on past data, that is,

in auction J ` 1 she chooses the reserve price prJ “ arg maxr ppprq. In expectation (over the

possible bids in the J`1th auction), this gives a profit of ppprJq, which, by definition, is lower

than the expected profit given by the optimal reserve price, ppr˚q. We study the size of this

difference, and how it changes as the number of observed auctions becomes large. We state

the bound in the following theorem. Its proof uses techniques from statistical learning to

probabilistically bound the difference between ppprq and pprq uniformly in r. Specifically,

the empirical Rademacher complexity (defined in the Appendix) plays a key role in obtaining

20Note that this approach to pick J can also be used to tackle the so-called “cold start” problem in onlineadaptive methods that also require that a number of absolute auctions be run such as Cesa-Bianchi et al.(2015).21A distinction between our work and Mohri and Medina (2016) is that we do not consider auction-levelcovariates, allowing us to derive an expression for the bound in terms of quantities that are known to (or canbe assumed by) the econometrician. Note that our goal here differs from that of Mohri and Medina (2016),who focused on improved algorithms to solve the optimization problem in (3) and (4).


this bound. These techniques are developed in Koltchinskii (2001) and Koltchinskii and

Panchenko (2002); Mohri et al. (2012) provide a textbook overview.

Theorem 4.

If Assumptions 1 and 3 are satisfied, for any δ ą 0, with probability at least 1´ δ over the possible

realizations of the J auctions, it holds that

ppr˚q´ppprJq

ω ď

˜

8?

log 2J ` 4

b

2`2 log JJ ` 6

c

log 4δ

2J

¸

.

To interpret this result, we can define ε as the gap between the profit at the estimated re-

serve price and the profit at the true optimal reserve price, normalized by the upper bound

of valuations, ω (and thus, ε P r0, 1s and can be interpreted as a fraction of the maximum

bidder valuation). Theorem 4 implies that, in order for the profit from the estimated re-

serve to be within ε of the profit from the optimal reserve with probability at least 1´ δ,

the seller needs to have observed approximately J auctions such that the expression in the

right-hand side of the expression in Theorem 4 equals ε. It can be shown that when J “ 1

the expression is positive and that it is strictly decreasing in J, which, in principle, enables

one to obtain the desired J for each pδ, εq via a simple nonlinear solver. Note also that the

right-hand side does not depend on any feature of the valuation distribution, and the the-

orem relates the optimal reserve, r˚, and the true expected profit function, pp¨q, without

requiring these objects to be known; it is due to these weak requirements that the bound is

conservative.

We illustrate the implications of Theorem 4 in Figure 1. For this illustration, we normal-

ize ω “ 1, and thus revenue is in units of fractions of the maximum willingness to pay.

We then plot “iso-data” curves in pδ, εq space, where each curve represents the possible

combinations of ε and δ that are possible given a fixed history of observed auctions. In

this figure, a curve located further to the southwest is preferable, as it represents a closer

approximation to the true optimal revenue (i.e. a smaller ε) with a higher probability (i.e.

a lower δ). The top line represents a sample size of J “ 1, 000, the middle line represents

J “ 5, 000, and the bottom line represents J “ 10, 000. The middle line suggests that with

a history of 5,000 auction realizations, one could guarantee a payoff within 0.348 (units of

the maximum willingness to pay) of the optimal profit with probability 0.975; or, with the

same size history, one could guarantee a payoff within 0.3447 of the optimal profit with

probability 0.70. The larger sample, J “ 10, 000, can guarantee a payoff that is much closer


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

J=10,000J=5,000J=1,000

FIGURE 1. Iso-data curves in pδ, εq spaceNotes: Figure displays combinations of pδ, εq that can be achieved given a fixed sample size J using thebound implied by Theorem 4. Top line represents J “ 1, 000, middle line represents J “ 5, 000, and bottomline represents J “ 10, 000.

to the optimal profit. Each iso-data curve is relatively flat in the δ dimension, reflecting the

fact that the sample size requirements are more stringent for achieving a given level of ε

closeness to the optimal profit, and are less stringent for achieving an improvement in δ

(i.e. in the probability with which the revenue is reached).

We now relate the explicit bound obtained in this subsection to the asymptotic results

obtained in the previous subsection. Theorem 2 implies that, for sufficiently large J, pprJq´

ppr˚q “ OppJ´2{3q.22 By definition, therefore, for any δ ą 0 and sufficiently large J, there

exists an M ą 0 such that PrpJ2{3|pprJq´ ppr˚q| ă Mq ě 1´ δ. The fact that the convergence

result in Theorem 2 is achieved at a J´2{3 rate implies that the bound in Theorem 4 is

conservative, as it is expressed as a function of plog J{Jq´1{2. However, Theorem 2 does not

allow one to explicitly compute the number of auctions required in order to approach the

optimal revenue with a given probability; it simply states that such an M exists if J is large

enough. The advantage of the bound in Theorem 4, on the other hand, is that it is explicit,

allowing one to directly compute an estimate (albeit a very conservative estimate) of the

22Performing a second-order Taylor expansion yields pprJq ´ ppr˚q “ p2prqprJ ´ r˚q2 “ OppJ´2{3q, where r isan intermediate value between rJ and r˚.


number of auctions J which must be observed for estimated reserve prices to perform well

without requiring knowledge of pp¨q or r˚.

In our application in Section 6, we take a step beyond asymptotic and learning the-

ory and demonstrate that each of these theoretical results may be quite conservative in

practice, as we find that, even with a relatively small data set of previously observed auc-

tions, revenue based on the estimated optimal reserve price can come quite close to the full

optimal-reserve-auction revenue.

5. MONTE CARLO SIMULATIONS

We now evaluate the finite-sample performance of our proposed procedure via Monte

Carlo simulations. We evaluate two different scenarios. In each scenario, we simulate

auctions with N “ 5 bidders with valuations drawn from a Ur0, 1s, and for simplicity

we set v0 “ 0. In the first scenario these valuations are independent and in the second

these valuations are perfectly correlated (with V1 “ ... “ V5).23 We report the difference in

expected profits from using the estimated reserve price using a small sample of historical

auction observations (samples of size J “ 10, 20, ..., 100 auctions) relative to the expected

profits from using the true optimal reserve price (computed based on a sample of 10,000

auctions). We use 500 replications of these simulations.

In addition to estimating reserve prices using our procedure, we also estimate reserve

prices by first estimating the full valuation distribution F. As highlighted in Section 2,

the only existing method for estimating the full distribution of valuations when only two

order statistics of bids are observed and N is unobserved is the approach of Song (2004).

This method relies on the novel insight that, in an iid environment, the distribution of a

higher order statistic conditional on a lower order statistic does not depend on N. Song

(2004) proposes using maximum likelihood estimation (MLE) to estimate F. The insight of

Song (2004) holds in a symmetric IPV environment. In such an environment, the optimal

reserve price can then be estimated in a number of possible ways once F is known. One

valid approach would be to choose r˚ as

arg maxr

rp1´ Fprqq. (6)

23We choose these two cases to evaluate our approach, but intermediate cases can also capture the relation-ships we document here.


The benefit of the Song (2004) method, in addition to providing an estimate of F, is that

it takes advantage of the strong assumption of symmetric IPV, and hence can yield more

efficient estimates of the reserve price than our direct approach. A drawback of the Song

(2004) approach is that it is not valid outside of symmetric IPV environment, unlike our

approach. Other drawbacks are that it requires approximating F (which our approach does

not), and that can be more costly in terms of computation time or ease of implementation

(our approach can be implemented with a simple grid search without requiring a numer-

ical optimization routine). We illustrate two approximations for F: a uniform distribution

Ur1, ws, where w is the parameter we estimate, and a fifth-degree Hermite polynomial

approximation for F. The former assumes much stronger knowledge about the shape of

the underlying valuation distribution (knowledge a researcher is unlikely to have in prac-

tice). Details on the estimation approach based on Song (2004) are found in the Online

Appendix.

Figures 2–4 displays the results of the simulation exercises. In each panel, solid lines

indicate the average across 500 Monte Carlo replications and dashed lines indicate asym-

metric 95% confidence intervals constructed from the 0.025 and 0.975 quantiles of the esti-

mator across the 500 replications; note that for some panels these confidence intervals are

so tight that they are indistinguishable from the solid lines. Panel A shows that, when data

truly is generated by a symmetric IPV process, the Song (2004) approach (the red lines,

“Est. full dist.”) outperforms our approach (the blue lines, “Directly est. r˚”) in terms of

expected loss. In panel C, when we use a Hermite polynomial approximation to the valu-

ation distribution instead of assuming knowledge that is a uniform distribution, the Song

(2004) approach still outperforms ours in terms of expected loss but to a lesser degree. The

performance of the directly estimated reserve prices improves as the sample size increases.

The real benefit of our approach is illustrated by panels B and D, where valuations arise

from a correlated PV environment. Here the assumptions of Song (2004) are not satisfied,

and the estimated reserve prices from that approach are clearly biased, with an average

expected loss around 24% of the true optimal reserve profit in panel B (using Uniform

MLE) and an average loss of about 3% in panel D (using Hermite MLE).

Figure 3 illustrates another strength of our approach, even in symmetric IPV environ-

ments: computation time. The average computation time for the Song (2004) approach is

a full 4 seconds in panel B, which uses the more computationally burdensome Hermite


FIGURE 2. Monte Carlo Simulations: Expected Loss

0 10 20 30 40 50 60 70 80 90 100

Number of historical auction observations

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Avg

. % lo

ss r

elat

ive

to tr

ue o

ptim

al r

eser

ve p

rofit Directly est. r*

Est. full dist.

(A) IPV, Uniform

0 10 20 30 40 50 60 70 80 90 100


-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Avg

. % lo

ss r

elat

ive

to tr

ue o

ptim

al r

eser

ve p


Est. full dist.

(B) Correlated PV, Uniform

0 10 20 30 40 50 60 70 80 90 100


-0.2

-0.1

0

0.1

0.2

0.3

Avg

. % lo

ss r

elat

ive

to tr

ue o

ptim

al r

eser

ve p


Est. full dist.

(C) IPV, Hermite

0 10 20 30 40 50 60 70 80 90 100


-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Avg

. % lo

ss r

elat

ive

to tr

ue o

ptim

al r

eser

ve p


Est. full dist.

(D) Correlated PV, HermiteNotes: Panel A shows average (across 500 Monte Carlo replications) of the expected loss from using reserveprices estimated from a sample of J “ 10, 20, ..., or 100 observations relative to true optimal reserve profit.“Directly use r˚” refers to reserve prices estimated using our approach, and “Est. full dist.” refers to resultsfrom estimating the full distribution of valuations using MLE following Song (2004) (using a uniformdistribution in panels A and B and using Hermite polynomials in panels C and D). Panels A and C show thesymmetric IPV setting and panels B and D the correlated PV setting. Asymmetric 95% confidence bands areshown with dashed lines.

approximation for F. In the correlated PV environment, our approach also performs faster

than the Song (2004) approach. In each panel of Figure 3 the average computation time for

the directly estimated reserve is less than 0.002 seconds.


FIGURE 3. Monte Carlo Simulations: Average Computation Time

0 10 20 30 40 50 60 70 80 90 100


-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Avg

tim

e (s

econ

ds)

of e

stim

atio

n

Directly est. r*

Est. full dist.

(A) IPV, Uniform

0 10 20 30 40 50 60 70 80 90 100


-0.04

-0.02

0

0.02

0.04

0.06

Avg

tim

e (s

econ

ds)

of e

stim

atio

n

Directly est. r*

Est. full dist.


0 10 20 30 40 50 60 70 80 90 100


-100

-50

0

50

100

150

Avg

tim

e (s

econ

ds)

of e

stim

atio

n

Directly est. r*

Est. full dist.

(C) IPV, Hermite

0 10 20 30 40 50 60 70 80 90 100


-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Avg

tim

e (s

econ

ds)

of e

stim

atio

n

Directly est. r*

Est. full dist.

(D) Correlated PV, HermiteNotes: Panel A shows average (across 500 Monte Carlo replications) computation time in seconds forestimated reserve prices using a sample of J “ 10, 20, ..., or 100 observations. “Directly use r˚” refers toreserve prices estimated using our approach, and “Est. full dist.” refers to results from estimating the fulldistribution of valuations using MLE following Song (2004) (using a uniform distribution in panels A and Band using Hermite polynomials in panels C and D). Panels A and C show the symmetric IPV setting andpanels B and D the correlated PV setting. Asymmetric 95% confidence bands are shown with dashed lines.

In Figure 4, panel A compares the expected loss from our approach to what could be

achieved if the econometrician were to observed all bids from the auction, yielding an im-

mediate estimate of F that can be plugged into (6) to obtain an estimate of r˚ in a symmet-

ric IPV environment. Panel A indicates that this kind of data would indeed be beneficial

relative to our estimation approach in a symmetric IPV setting. Panel B demonstrates,

however, that this approach would be biased in a correlated PV setting. This is because,

in a general correlated private value function, all information about the seller’s profit (and


hence the optimal reserve price) is contained in the marginal distributions of the first- and

second-highest bids; lower order statistics of bids contain no additional information for

the seller’s profit optimization problem.24

We also highlight here that observing all bids is not possible in the sequential-arrival

auction we model, where some bidders can arrive after the current bid has passed their

valuation. Panel C of Figure 4 demonstrates that erroneously using the observed bids (used

for the estimates show in green) as though they represent all bids (used for the estimates

shown in purple) will lead to biased reserve prices, even in a symmetric IPV environment.

Panel D demonstrates the sensitivity of our approach to Assumption 2, which requires

that the data generating process for would-be bids be the same the historical auctions and

the future auctions. In panel D the blue line indicates results using our approach and the

yellow line indicates the expected loss when we instead artificially inflate historical bids

by 1%, holding fixed the true distribution of bids. These inflated historical bids lead to a

clear bias in estimated reserve prices that does not disappear as the sample size increases.

6. COMPUTING OPTIMAL RESERVE PRICES IN E-COMMERCE AUCTIONS

We apply our methodology to a dataset of eBay auctions selling commodity-like prod-

ucts, which we define as those products which are cataloged in one of several commer-

cially available product catalogs. Examples of commodity products include ”Microsoft

Xbox One, 500 GB Black Console”, ”Chanel No.5 3.4oz Women’s Eau de Parfum”, and

”The Sopranos - The Complete Series (DVD, 2009)”. We will refer to each distinct product

as a “product” or “product-category.” Within each product, the items sold are relatively

homogeneous. For this exercise, we select popular iPhone products listed through auc-

tions from 2011–2015. We consider only auctions with no reserve price; specifically, we

only include auctions for which the start price was less than or equal to $0.99, the default

start price recommendation on eBay. We omit auctions in which the highest bid is in the

top 1% of all highest bids for that product and limit to products that are auctioned at least

1,000 times in our sample.

24Athey and Haile (2007) and Aradillas-Lopez et al. (2013) discuss how, in the general correlated private valuescase, the seller gains no additional benefit in computing reserve prices by observing additional bids other thanthe two highest.


FIGURE 4. Monte Carlo Simulations: All Bids vs. Observed Bids vs. Biased Bids

0 10 20 30 40 50 60 70 80 90 100


-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Avg

. % lo

ss r

elat

ive

to tr

ue o

ptim

al r

eser

ve p


Use all bids

(A) IPV

0 10 20 30 40 50 60 70 80 90 100


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

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. % lo

ss r

elat

ive

to tr

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al r

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ve p


Use all bids


0 10 20 30 40 50 60 70 80 90 100


-0.06

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-0.02

0

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0.04

0.06

Avg

. % lo

ss r

elat

ive

to tr

ue o

ptim

al r

eser

ve p

rofit Use all bids

Use obs. bids

(C) IPV

0 10 20 30 40 50 60 70 80 90 100


-0.6

-0.4

-0.2

0

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0.6

Avg

. % lo

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ve p

rofit Correct bids

Biased bids

(D) IPVNotes: Panel A shows average (across 500 Monte Carlo replications) of the expected loss from using reserveprices estimated from a sample of J “ 10, 20, ..., or 100 observations relative to true optimal reserve profit.“Directly use r˚” (and “Correct bids” in panel D) refers to reserve prices estimated using our approach; “Useall bids” refers to results from estimating the full distribution of valuations from all bids; “Use obs. bids”refers to results from estimating the full distribution of valuations from only those bids that are not censoredby bidders’ sequential arrival; “Biased bids” in panel D refers to results using bids historical bids that are 1%higher than the true simulated draws. Panels A, C, and D show the symmetric IPV setting and panel B thecorrelated PV setting. Asymmetric 95% confidence bands are shown with dashed lines.

Table 1 shows summary statistics at the product level. There are 22 distinct iPhone

products in our sample, with the number of auctions per product ranging from 1,144 to

11,733. Table 1 shows evidence of price dispersion within a product across auctions. Our

modeling framework rationalizes such dispersion through different numbers of bidders

arriving to different auctions—more bidders will result in a higher auction price—and


TABLE 1. Product-Level Descriptive Statistics

Highest Bid Second Highest Bid

Product # Obs Average($) Std Dev ($) Average ($) Std Dev ($)

1. iP3GS, 16GB, AT&T 3,907 147.42 68.50 135.00 67.032. iP3GS, 32GB, AT&T 1,197 170.72 70.62 155.51 69.113. iP3GS, 8GB, AT&T 2,705 97.28 55.29 86.62 53.004. iP3G, 8GB, AT&T 3,545 95.43 47.95 85.16 45.865. iP4S, 16GB, AT&T 5,761 243.94 130.22 221.95 121.396. iP4S, 16GB, Sprint 2,978 205.74 110.19 185.80 104.087. iP4S, 16GB, Unlocked 1,199 297.08 148.26 267.59 124.258. iP4S, 16GB, Verizon 4,096 211.89 115.28 190.95 109.049. iP4S, 32GB, AT&T 1,493 292.21 133.64 266.66 128.5710. iP4, 16GB, AT&T 11,733 231.28 105.86 213.13 101.2411. iP4, 16GB, Unlocked 1,590 258.09 121.76 235.89 110.2212. iP4, 16GB, Verizon 6,698 161.37 89.05 145.96 85.3913. iP4, 32GB, AT&T 4,245 259.57 110.11 239.14 107.5014. iP4, 32GB, Verizon 1,600 187.68 97.59 169.75 94.4715. iP4, 8GB, AT&T 2,302 150.01 86.51 134.80 80.8416. iP4, 8GB, Sprint 2,198 116.88 69.30 103.19 66.1617. iP4, 8GB, Verizon 3,003 112.51 69.17 99.70 64.8418. iP5, 16GB, AT&T 1,553 348.83 179.91 311.52 155.6019. iP5, 16GB, T-Mobile 1,284 267.09 175.58 235.52 145.1420. iP5, 16GB, Unlocked 3,381 288.81 164.58 257.38 149.7021. iP5, 16GB, Verizon 1,605 307.24 152.28 271.54 132.9522. iP5, 64GB, AT&T 1,144 231.86 133.43 206.87 121.88

Notes: Table displays, for each product, the number of auctions recorded and the average and standarddeviation of the first and second highest bids.

different realizations of bidders’ valuations in different auctions. Given that reserve prices

increase revenue only when they lie between the highest and second highest bids, the size

of the gap between these bids is of particular interest. This gap ranges from $10.27 (12% of

the mean second highest bid) for product 4 to $37 for product 18 (also 12% of this product’s

the mean second highest bid). This gap is not large, suggesting that these products may

have relatively competitive markets on eBay and that reserve prices may only be able to

increase revenue slightly over a no-reserve auction for these products.

We estimate optimal reserves separately in each of the 22 products. Figures 5 displays

the gain from using the optimal reserve price, where we evaluate these gains in three

different ways. In Panel A we consider the case where v0 “ 0 and we report the percentage

gain in revenue from using the estimated optimal reserve price relative to an auction with


no reserve price (or a reserve price of zero). In panel A, we find that for v0 “ 0 it is not clear

whether any positive reserve is beneficial: the expected percentage gain is less than 0.01 for

each product and not statistically significantly different from zero for most products. Note

that treating v0 “ 0 (regardless of the seller’s true valuation) will yield the reserve price

that will maximize the expected payment of the winning bidder to the seller, which may

be the quantity that the auction platform (here, eBay) is most interested in maximizing, as

platform fees are typically proportional to this payment. Our results in panel A therefore

suggest that eBay would prefer a zero reserve for these products, consistent with eBay’s

practice of recommending low reserve prices (0.99) to sellers.

In panel B we set v0 “ ErVp2qs and consider the expected gain from using the optimal

reserve price relative to an auction with r “ v0. We consider the r “ v0 auction as our

benchmark because the seller would clearly find it suboptimal to sell the good at a price

lower than her valuation v0. Our specific choice of setting v0 to the expected second-

highest bid is a form of capturing the seller’s perceived value of keeping the good herself.

It also captures in a simple way what the seller might expect from re-auctioning the item.

Here we find again small gains from implementing the optimal reserve price (less than

0.01), but the gains are statistically significantly different from zero for most products.

In panel C we consider the case where the seller re-auctions the item if it fails to sell,

using Equation (5) with a discount factor of β “ 0.9.25 We evaluate the gain from using the

optimal reserve price in this scenario, and here the gains are large and significant, ranging

from 22% to 44%. These gains are larger than in panels A and B because here we evaluate

the gains relative to a scenario in which the seller uses no reserve price. The reason for this

choice of benchmark is the following. Unlike in the static auction case, where a benchmark

reserve price related to the seller’s valuation, v0, is natural, there is no role for v0 in the

infinite-horizon, repeated-auction case. Rather, the seller’s outside option when a given

auction fails is defined recursively as the option to relist the item. Any arbitrary choice of

an alternative, sub-optimal reserve price would also work as a benchmark, but we see the

no-reserve benchmark as the most reasonable choice for this case.

25This choice of β is only for illustrative purposes, and it implies that the seller discounts the future by morethan would be implied by the time value of money alone; we consider this discount factor as also capturingother unmodeled features that may result in impatience on the seller’s part or that may prohibit the sellerfrom quickly re-auctioning the item. The results can easily be generated with other values of β.


FIGURE 5. Revenue Increase from Optimal Reserve Price

22212019181716151413121110

987654321

Prod

uct

-.005 0 .005 .01Percentage Increase

(A) v0 “ 0

22212019181716151413121110

987654321

Prod

uct

-.004 -.002 0 .002 .004 .006Percentage Increase

(B) v0 “ EpVp2qq

22212019181716151413121110

987654321

Prod

uct

.2 .25 .3 .35 .4 .45Percentage Increase

(C) Repeated AuctionNotes: Expected revenue increase from using estimated optimal reserve price relative to using no reserveprice in panel A (where v0 “ 0) and relative to a reserve price of r “ ErVp2qs in panel B (where v0 “ ErVp2qs)and relative to no reserve price in panel C, where the seller’s outside option is to re-auction the item. 95%confidence interval is shown in red, based on 1,000 subsample draws of size 250 from the full sample,separately for each product.


In Figure 6, we select one specific product in our sample, product #17, which is the 8GB

version of the Apple iPhone 4 locked to Verizon. In the figure we plot expected profit as a

function of the reserve price given the empirical distribution of the first and second highest

bids. Panel A considers the setting where v0 “ 0, panel B considers the setting where v0 is

the average second order statistic, and panel C consider the repeated auction case. Unsur-

prisingly for a product supplied elastically on other online or offline platforms, the figure

shows that there is a sharp drop-off in profit for reserves beyond a certain point. This large

drop off illustrates a point also discussed in Ostrovsky and Schwarz (2016) and Kim (2013):

the loss from setting a non-optimal reserve price is asymmetric, such that overshooting the

optimal reserve results in a much larger loss in magnitude than undershooting it. The ver-

tical line represents the estimated optimal reserve price.

We now turn to the question of how close optimal reserve prices will be to those esti-

mated using a finite history of first and second-highest bids. The theoretical guarantee of

Theorem 4 assures us that estimated reserve prices will eventually perform close to opti-

mally. We assess this feature through a simulation exercise.

For each product, we draw 1,000 sequences, each of length 250, at random with replace-

ment from the empirical distribution of all auctions observed for that product over the

sample period. Within each sequence, we then estimate the reserve price suggested by our

approach using only the first τ observations in the sequence, doing so separately for each

τ P t2, ..., 250u. Thus, we begin with only 2 historical auction observation, then 3, then

4, and so on, for each drawn sequence. Next, at each of these estimated reserve prices,

using the full sample of historical observations for the product, we compute the expected

profit the seller would receive from using this computed reserve price. Therefore, for this

exercise we treat the empirical distribution of auctions in our sample as representing the

“true” distribution of first and second highest bids, and we treat sellers as only having in-

formation on a history of τ auctions drawn at random from the full empirical distribution.

Figure 7 shows the results of this exercise for the same product as in Figure 6. In panels

A, C, and E, the quantities on the y-axis are expressed as the expected loss in profit from

using reserve prices estimated from a given small sample size relative to the profit from

using the “true” optimal reserve price. The solid line represents the average loss, averaged

across the 1,000 subsample draws, and the dashed lines represent pointwise 95% confi-

dence intervals (the 0.025 and 0.975 quantiles from the simulations). In panels B, D, and F


FIGURE 6. Profit Under Different Reserve Prices for Apple iPhone 4 8GB Verizon

0 20 40 60 80 100 120 140 160 180 200

Reserve Price

20

30

40

50

60

70

80

90

100

Rev

enue

, iP

hone

4, 8

GB

, Ver

izon

(A) v0 “ 0

0 50 100 150 200 250 300 350 400 450

Reserve Price

95

100

105

110

115

120

125

130

Rev

enue

, iP

hone

4, 8

GB

, Ver

izon

(B) v0 “ EpVp2qq

0 50 100 150 200 250 300 350 400 450 500

Reserve Price

-20

0

20

40

60

80

100

120

140

160

Rev

enue

, iP

hone

4, 8

GB

, Ver

izon

(C) Repeated Auction

Notes: Seller expected profit as a function of the reserve price, given the empirical distribution of first andsecond highest bids, for Apple iPhone 4 8GB Verizon (using all observations for this product). Panel A setsv0 “ 0, panel B sets v0 to the average second highest bid, and panel C considers a repeated auction, where theoutside option is to re-auction the item. Vertical line displays location of optimal reserve price.

we plot the actual estimated reserve price from each of these samples. Thus, in each panel,

the x-axis represents the sample size used to compute the optimal reserve price (for panels

on the right) and the corresponding loss in profit from using this reserve price (for panels

on the left).

As expected given Theorem 4, the loss does indeed converge to zero (i.e. the profit

converges to the true optimal reserve profit level). In the initial phases, estimated reserve

prices can be seriously suboptimal, even compared to setting a reserve of r “ v0. However,


FIGURE 7. Expected Loss and Average Reserves From Computation withDifferent Numbers of Observed Auctions for Apple iPhone 4 8GB Verizon

0 50 100 150 200 250

Number of Auction Observations

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Exp

ecte

d Lo

ss, i

Pho

ne 4

, 8G

B, V

eriz

on

(A) Exp. Loss, v0 “ 0

0 50 100 150 200 250


0

50

100

150

200

250

Res

erve

Pric

e, iP

hone

4, 8

GB

, Ver

izon

(B) Avg. Reserve, v0 “ 0

0 50 100 150 200 250


-0.02

0

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0.14

0.16

0.18

Exp

ecte

d Lo

ss, i

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, 8G

B, V

eriz

on

(C) Exp. Loss, v0 “ EpVp2qq

0 50 100 150 200 250


80

100

120

140

160

180

200

220

240

260

280

Res

erve

Pric

e, iP

hone

4, 8

GB

, Ver

izon

(D) Avg. Reserve, v0 “ EpVp2qq

0 50 100 150 200 250


-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Exp

ecte

d Lo

ss, i

Pho

ne 4

, 8G

B, V

eriz

on

(E) Exp. Loss, Repeated Auc.

0 50 100 150 200 250


0

50

100

150

200

250

300

350

Res

erve

Pric

e, iP

hone

4, 8

GB

, Ver

izon

(F) Avg. Reserve, Repeated Auc.Notes: Panels on left show expected loss (as a fraction of the true optimal expected profit) from using estimated reserveprices as a function of the number of auctions observed, where simulations are conducted by drawing sequences ofauctions from the empirical distribution of first and second highest bids from auctions for Apple iPhone 4 8GB Verizon andcomputing the estimated expected profit progressively adding each auction one at a time. Panels on the right show theaverage estimated optimal reserve price. Solid line represents the average across the 1,000 simulation replications anddashed lines represent 95% confidence intervals. Panels A and B set v0 “ 0, panels B and C set v0 to the average secondhighest bid, and panels D and E consider a repeated auction, where the outside option is to re-auction the item.


convergence to the optimal level appears to occur quite quickly. The optimal reserve prices

themselves also converge relatively quickly. Initially, with especially small sample sizes,

the estimated reserves are, on average, too high. The lower confidence bands in panels

B, D, and F demonstrate that the estimated reserve prices can also be too low in some

samples. These confidence bands shrink quickly as the sample size grows. This quick

convergence to the truth is quite robust regardless of what outside option the seller chooses

to consider. Figures corresponding to Figures 6 and 7 for all products are found in the

Online Appendix.

We now address the question of how many auction observations a practitioner may wish

to collect before implementing the estimated optimal reserve price. Table 2 displays results

using the same simulation exercise described above, evaluated separately for each prod-

uct. The first three columns show (for our three scenarios for the seller’s outside option)

the median number of historical auctions (across the 1,000 simulation draws) required to

achieve a revenue level that is within 1% of the true optimal-reserve-auction revenue. For

the v0 “ 0 case, this ranges from 6–10 auctions. For the v0 “ ErVp2qs case, as few as 1–2

auctions can achieve a revenue within 1% of the truth. For the repeated auction case, 4–22

auctions are required.

The last three columns of Table 2 address this question differently. We consider a sce-

nario in which a seller has an inventory of 250 iPhones to sell, and she wishes to run J˚

no-reserve auctions to collect data, and then in the remaining p250´ J˚q auctions she will

implement the optimal reserve price estimated from the first J˚ auctions. The seller solves

J˚ “ arg maxJ

Jpp0q ` p250´ JqpprJq. (7)

This problem is not actually feasible in practice because it depends on knowing the true

profit function pp¨q.26 Nonetheless, we consider it an interesting thought experiment. We

report the resulting J˚ from solving (7) separately for each product in the last three columns

of Table 2. Here we find quantities that are of a similar order of magnitude to those in

the first three columns, suggesting again that estimating and implementing a reserve price

from even small samples (less than 25) of historical auctions can be profitable. These results

also suggest that, if a mechanism designer concerned with changes in demand wishes to

26The results of this exercise also depend crucially on the size of the inventory; here, 250. A different inventorysize would results in a different data-collection threshold.


TABLE 2. How Many No-Reserve Auctions to Run?

Within 1% Opt. Rev. Maximize Rev. 250 Sales

v0 “ 0 v0 “ ErVp2qs Repeat v0 “ 0 v0 “ ErVp2qs Repeat1. iP3GS, 16GB, AT&T 8 1 5 12 1 122. iP3GS, 32GB, AT&T 10 1 4 10 1 123. iP3GS, 8GB, AT&T 6 1 8 11 2 184. iP3G, 8GB, AT&T 7 1 9 11 1 155. iP4S, 16GB, AT&T 8 1 7 10 1 246. iP4S, 16GB, Sprint 9 1 8 10 1 187. iP4S, 16GB, Unlocked 7 1 5 10 1 18. iP4S, 16GB, Verizon 8 1 9 11 1 219. iP4S, 32GB, AT&T 7 1 7 11 1 2010. iP4, 16GB, AT&T 7 2 4 10 2 111. iP4, 16GB, Unlocked 8 1 5 10 1 112. iP4, 16GB, Verizon 7 1 15 10 1 1613. iP4, 32GB, AT&T 9 2 5 12 1 1814. iP4, 32GB, Verizon 8 1 11 10 1 1615. iP4, 8GB, AT&T 6 1 9 11 1 2116. iP4, 8GB, Sprint 7 1 16 10 1 1817. iP4, 8GB, Verizon 8 1 10 11 1 218. iP5, 16GB, AT&T 7 1 7 11 1 119. iP5, 16GB, T-Mobile 7 1 10 11 1 220. iP5, 16GB, Unlocked 7 1 8 9 1 221. iP5, 16GB, Verizon 7 1 8 10 1 122. iP5, 64GB, AT&T 9 1 22 11 1 2

Notes: For each product, the first three columns displays (for the three different seller outside optionscenarios) the median number of auctions required for the computed reserve price to yield an expected profitthat is within 1% of the true optimal reserve auction profit. The median is taken across 1, 000 simulatedsamples drawn with replacement from the full set of observations for that product. The last three columnsdisplay the optimal number of no-reserve auctions to run prior to estimating and implementing the optimalreserve on the remainder of the auctions in the inventory, where the inventory size in this exercise is 250items.

“reset” or “update” the optimal reserve price based on recently observed auctions, doing

so may not be very costly, as each instance of setting a new reserve may only require a

small sample of recent auctions.


7. CONCLUSION

We study a computationally simple approach for estimating optimal reserve prices in

asymmetric, correlated private values settings. The approach applies to settings with in-

complete bidding data where only the top two bids are observed and where the number

of bidders is unknown. These data requirements are frequently met in online (advertis-

ing or e-commerce) settings. We also derive a bound on the number of auction records

one needs to observe in order for realized revenue based on estimated reserve prices to

approximate the optimal revenue closely. We illustrate the approach using eBay auctions

of used iPhones, and illustrate that revenue could potentially increase if optimal reserve

prices were employed in practice. We examine the empirical relevance of our theoretical

results and find that fewer than 25 auctions need to be recorded prior to estimating re-

serve prices in order for the estimated reserve price to yield an expected loss of less than

1% relative to the true optimal reserve revenue.

While the approach abstracts away from a number of information settings or real-world

details (such as common values or inter-auction dynamics), we believe the virtue of the

approach is its simplicity, providing a tractable and scalable approach to computing re-

serve prices even in large, unwieldy datasets where typical computationally demanding

empirical auction approaches would be infeasible.


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APPENDIX A. PROOFS

Proof of Theorem 1

Proof. We will show that pprq converges uniformly in probability to pprq. Note that

πpVp1qj , Vp2qj , v0, rq can alternatively be written as maxtr, Vp2qj u ` pv0 ´ rq1pVp1qj ă rq (see,

for example, Aradillas-Lopez et al. 2013). For simplicity, assume that v0 “ 0 and let

p1prq “ 1JřJ

j“1 max!

r, Vp2qj

)

and p2prq “ ´1JřJ

j“1 r1!

Vp1qj ă r)

, so that pprq “ p1prq `

p2prq. Notice that p1prq is Lipschitz continuous because, for any r1 and r2, it follows that

|p1pr1q ´ p1pr2q| ď |r1 ´ r2|. Furthermore, for any r, it follows by the law of large numbers

that p1prqpÝÑ p1prq. Thus, we can invoke Lemma 2.9 in Newey and McFadden (1994) to

obtain suprPR |p1prq ´ p1prq|pÝÑ 0. Finally, it is straightforward to check that the function

f px, rq “ ´r1tx ď ru belongs to a VC subgraph class (see, for example, van der Vaart and

Wellner 1996), which guarantees uniform convergence of p2p¨q. Consequently, we have

suprPR |pprq ´ pprq|pÝÑ 0, which guarantees that rJ

pÝÑ r˚ and pprJq

pÝÑ ppr˚q. �

Proof of Theorem 2

Proof. We first note that the Kim and Pollard (1990) result requires the following assump-

tion, the definition of which is somewhat involved. Let πp¨, rq ” πp¨, rq ´ πp¨, r˚q and let

PRp¨q be defined as the supremum of |πp¨, rq| over the class PR ” tπp¨, rq : |r ´ r˚| ď Ru.

The functions PRp¨q are referred to as envelopes of the classes PR. The class PR is referred

to as manageable for the envelope PR (Def. 4.1 of Pollard 1989) if there exists a decreasing

function Dp¨q for which (i)ş1

0plog Dpxqq1{2dx ă 8 and (ii) for every measure Q with finite

support, D2pεpQP2Rq

1{2,PR, Qq ă Dpεq for 0 ă ε ă 1, where the term D2pε,PR, Qq equals

the largest J for which there are functions PR,1, ..., PR,J in PR with Prj |PR,k ´ PR,j|2 ą ε2 for

k ‰ j; and where Prj is the empirical probability measure. The class PR is then referred

to as uniformly manageable if there exists a function that bounds every subclass of PR; the

precise properties this bounding function must satisfy are described in detail in Section 3

of Kim and Pollard (1990). The condition required by Kim and Pollard (1990), applied to

our context, is as follows:

Assumption 4.

The classes PR, for R near 0, are uniformly manageable for the envelopes PR.


It is straightforward, though tedious, to show that this assumption of uniformly man-

ageability is satisfied in our context. We omit the proof here.

Throughout the rest of the proof of our Theorem 2, we use r1 and r2 such that, without

loss of generality, r1 ą r2 ą r˚. We also denote the marginal densities of Vp1q and Vp2q

by f1p¨q and f2p¨q, respectively. Also, let ´Σ denote the second derivative of Erπp¨, rqs

evaluated at r˚.

By the main theorem of Kim and Pollard (1990), if Assumptions 3 and 4 are satisfied,

and r˚ is an interior point, and if the following conditions hold

(1) Hpβ, αq “ limtÑ01t Erπp¨, r˚ ` βtqπp¨, r˚ ` αtqs exists for each β, α in R and

limtÑ0

1t

Erπp¨, r˚ ` αtq21t|πp¨, r˚ ` αtq| ą ε{tus “ 0

for each ε ą 0 and α in R;

(2) ErP2Rs “ OpRq as R Ñ 0 and for each ε ą 0 there is a constant K such that

ErP2R1tPR ą Kus ă εR for R near 0;

(3) Er|πp¨, r1q ´ πp¨, r2q|s “ Op|r1 ´ r2|q near r˚;

then the process J2{3 1JřJ

j“1 πp¨, r˚ ` αJ´1{3q converges in distribution to a Gaussian pro-

cess Zpαqwith continuous sample paths, expected value ´ 12 α2Σ, and covariance kernel H,

where Hpβ, αq “ limtÑ01t Erπp¨, r˚ ` βtqπp¨, r˚ ` αtqs for any β, α in R. Furthermore, if Z

has nondegenerate increments, then J1{3prJ ´ r˚q converges in distribution to the random

maximizer of Z. We now prove that conditions (1)–(3) above are satisfied.

To establish condition (1), we first characterize the limiting behavior of 1t Erπp¨, r˚ `

βtqπp¨, r˚` αtqs as t Ñ 0. By the definition of πp¨, rq, this amounts to studying the behavior

of four different terms, which we conduct separately below. Let r1 “ r˚ ` αt and r2 “

r˚ ` βt. First, we consider the term

h1 ”

´

maxtVp2q, r1u ´maxtVp2q, r˚u¯´

maxtVp2q, r2u ´maxtVp2q, r˚u¯

.

Notice that when Vp2q ą r2 then h1 “ 0; when Vp2q ă r˚ then h1 “ pr1 ´ r˚qpr2 ´ r˚q; and

when r˚ ă Vp2q ă r2 then h1 “ pr1 ´Vp2qqpr2 ´Vp2qq. Therefore,

1t

E”´

maxtVp2q, r1u ´maxtVp2q, r˚u¯´

maxtVp2q, r2u ´maxtVp2q, r˚u¯ı

“1t

!

pr1 ´ r˚qpr2 ´ r˚qPrpVp2q ă r˚q


`E”

pr1 ´Vp2qqpr2 ´Vp2qq|r˚ ă Vp2q ă r2

ı

Prpr˚ ă Vp2q ă r2q

)

“1t

#

αβt2ż r˚

0f2puqdu` r1r2

ż r2

r˚

f2puqdu´ pr1 ` r2q

ż r2

r˚

u f2puqdu`ż r2

r˚

u2 f2puqdu

+

“1t

αβt2F2pr˚q `“

pr˚q2 ` pα` βqt` αβt2‰ r f2pr˚qpr2 ´ r˚q ` opr2 ´ r˚qs

´ r2r˚ ` pα` βqts rr˚ f2pr˚qpr2 ´ r˚q ` opr2 ´ r˚qs `“

pr˚q2 f2pr˚qpr2 ´ r˚q ` opr2 ´ r˚q‰(

“1t

αβt2F2pr˚q ` αβt2 f2pr˚qβt` optq(

“1t

optq “ op1q.

The second term we consider is

h2 ”

´

r˚1tVp1q ă r˚u ´ r21tVp1q ă r2u

¯´


.

When Vp1q ă r˚, h2 “ pr˚ ´ r2qpr1 ´ r˚q; when r˚ ă Vp1q ă r2 and Vp2q ă r˚, h2 “

´r2pr1 ´ r˚q; and when r˚ ă Vp1q ă r2 and r˚ ă Vp2q ă Vp1q, h2 “ ´r2pr1 ´Vp2qq. Hence,

1t

E”´


¯´


“1t

!

pr˚ ´ r2qpr1 ´ r˚qPrpVp1q ă r˚q ´ r2pr1 ´ r˚qPrpVp2q ă r˚ ă Vp1q ă r2q

´r2r1Prpr˚ ă Vp2q ă Vp1q ă r2q

`ErVp2q|r˚ ă Vp2q ă Vp1q ă r2sPrpr˚ ă Vp2q ă Vp1q ă r2q

)

“1t

#

´αβt2F1pr˚q ´ r2αtż r2

r˚

ż r˚

0f1,2pu, vqdudv´ r1r2

ż r2

r˚

ż v

r˚

f1,2pu, vqdudv

`r2

ż r2

r˚

ż v

r˚

u f1,2pu, vqdudv*

“1t

"

´αβt2F1pr˚q ´ r2αtż r2

r˚

Fvpr˚, vqdv´ r1r2

ż r2

r˚

rFvpv, vq ´ Fvpr˚, vqsdv

`r2

ż r2

r˚

rF1,2pv, vq ´ F1,2pv, r˚qsdv*

“1t

´αβt2F1pr˚q ´ r2αtrβtFvpr˚, r˚q ` optqs ´ r1r2optq ` r2optq(

“1t

optq “ op1q.


The third term is

h3 ”

´


¯´


.

The term h3 “ 0 when Vp1q ą r1 or Vp2q ą r2; h3 “ ´r1pr2´ r˚qwhen Vp2q ă r˚ ă Vp1q ă r1;

h3 “ ´r1pr2 ´Vp2qq when r˚ ă Vp1q ă r1 and r˚ ă Vp2q ă r2; and h3 “ pr˚ ´ r1qpr2 ´ r˚q

when Vp1q ă r˚. Consequently,

1t

E”´


¯´


“1t

!

´r1pr2 ´ r˚qPrpVp2q ă r˚ ă Vp1q ă r1q

´r1

´

r2 ´ErVp2q|r˚ ă Vp1q ă r1, r˚ ă Vp2q ă r2s

¯

Prpr˚ ă Vp1q ă r1, r˚ ă Vp2q ă r2q

)

“1t

#

´r1βtż r1

r˚

ż r˚

0f1,2pu, vqdudv´ αβt2F1pr˚q ´ r1r2

ż r2

r˚

ż v

r˚

f1,2pu, vqdudv

´r1r2

ż r1

r2

ż r2

r˚

f1,2pu, vqdudv

`r1

ż r2

r˚

ż v

r˚

u f1,2pu, vqdudv` r1

ż r1

r2

ż r2

r˚

u f1,2pu, vqdudv*

“1t

"

´r1βtż r1

r˚

Fvpr˚, vqdv´ αβt2F1pr˚q ´ r1r2

ż r2

r˚

rFvpv, vq ´ Fvpr˚, vqsdv

´r1r2

ż r1

r2

f1,2pr˚, vqpr2 ´ r˚qdv

`r1

ż r2

r˚

rF1,2pv, vq ´ F1,2pv, r˚qsdv` r1r˚ż r1

r2

f1,2pr˚, vqpr2 ´ r˚qdv` orpr2 ´ r˚q2s*

“1t

´r1αβt2Fvpr˚, r˚q ´ αβt2F1pr˚q ´ r1r2opr1 ´ r˚q

`r1opr2 ´ r˚q ´ r1β2t2ż r1

r2

f1,2pr˚, vqdv` orpr2 ´ r˚q2s*

“1t

optq “ op1q.

The fourth and final term is

h4 ”

´


¯´


¯

.


The term h4 “ 0 if Vp1q ą r2; h4 “ r1r2 if r˚ ă Vp1q ă r2; and h4 “ pr˚ ´ r2qpr˚ ´ r1q if

Vp1q ă r˚. Thus,

1t

E”´


¯´


¯ı

“1t

!

r1r2Prpr˚ ă Vp1q ă r2q ` pr˚ ´ r2qpr˚ ´ r1qPrpVp1q ă r˚q)

“1t

"

r1r2

ż r2

r˚

f1pvqdv` αβt2F1pr˚q*

“1t

r1r2r f1pr˚qpr2 ´ r˚q ` opr2 ´ r˚qs ` αβt2F1pr˚q(

1t

pr˚q2 f1pr˚qβt` optq(

“ pr˚q2 f1pr˚qβ` op1q.

These four results show that the limit Hpβ, αq is well defined for each β, α in R, which es-

tablishes the first part of (1). For the second part of (1), notice that |πp¨, r˚` αtq| is bounded

for any t, which means that there exists a t ă 8 such that the indicator would be 0 for all

t ă t, establishing the result.

We now establish (2). Let R ą 0 and r be the maximizer of πp¨, rq such that |r´ r˚| ă R.

We first need to show that Erπp¨, rq2s “ OpRq. We will split the analysis into three terms.

First, notice that for the first term:

pmaxtVp2q, ru ´maxtVp2q, r˚uq2 “ p|maxtVp2q, ru ´maxtVp2q, r˚u|q2 ď p|r´ r˚|q2 ă R2

which implies that its expected value is OpR2q. Moving to the next two terms we will

assume that r ą r˚, as the calculations for the opposite case are analogous. For the second

one,

pr˚1tVp1q ă r˚u ´ r1tVp1q ă ruq2 “ pr˚q21tVp1q ă r˚u ` r21tVp1q ă ruq2 ´ 2r˚r1tVp1q ă r˚u

so taking expectations yields:

pr˚q2ż r˚

0f1pvqdv´ r˚r

ż r˚

0f1pvqdv``r2

ż r

r˚

f1pvqdv` r2ż r˚

0f1pvqdv´ r˚r

ż r˚

0f1pvqdv

“ pr´ r˚q2F1pr˚q ` r2r f1pr˚qpr´ r˚q ` opr´ r˚qs

“ Opr´ r˚q ` opr´ r˚q “ Opr´ r˚q ă OpRq.


The third and last term is given by:

E”

pmaxtVp2q, ru ´maxtVp2q, r˚uqpr˚1tVp1q ă r˚u ´ r1tVp1q ă ruqı

“ E”

pmaxtVp2q, r˚ur˚1tVp1q ă r˚uı

É”

pmaxtVp2q, rur1tVp1q ă ruı

È”

pmaxtVp2q, r˚ur1tVp1q ă ruı

É”

pmaxtVp2q, rur˚1tVp1q ă r˚uı

“ pr˚q2ż r˚

0

ż v

0f1,2pu, vqdudv´ r2

ż r

0

ż v

0f1,2pu, vqdudv

` rż r

0

ż v

0maxtu, r˚u f1,2pu, vqdudv´ r˚r

ż r˚

0

ż v

0f1,2pu, vqdudv

“ rpr˚q2 ´ r2s

ż r˚

0

ż v

0f1,2pu, vqdudv´ r2

ż r

r˚

ż v

0f1,2pu, vqdudv

` rż r

r˚

ż v

0maxtu, r˚u f1,2pu, vqdudv

“ pr˚ ´ rqpr˚ ` rqF1,2pr˚, r˚q ´ r2ż r

r˚

Fvpv, vqdv` r˚rż r

r˚

ż r˚

0f1,2pu, vqdudv

` rż r

r˚

ż v

r˚

u f1,2pu, vqdudv

“ Opr˚ ´ rq ´ rpr´ r˚qrFvpr˚, r˚qpr˚ ´ rq ` opr˚ ´ rqs ` rż r

r˚

Gpvqdv

“ Opr˚ ´ rq ` rrGpr˚qpr´ r˚q ` opr´ r˚qs “ Opr´ r˚q ă OpRq,

which, along with the results for the previous two terms, establishes the first part of condi-

tion (2). The second part of condition (2) follows directly from the integrability of πp¨, rq2.

To verify that (3) holds, notice that:

|πpξ j, r1q ´ πpξ j, r2q|

“ |πpξ j, r1q ´ πpξ j, r2q|

“

ˇ

ˇ

ˇmax

!

Vp2qj , r1

)

´ r11

!

Vp1qj ă r1

)

´max!

Vp2qj , r2

)

` r21

!

Vp1qj ă r2

)ˇ

ˇ

ˇ

ď

ˇ

ˇ

ˇmax

!

Vp2qj , r1

)

´max!

Vp2qj , r2

)ˇ

ˇ

ˇ`

ˇ

ˇ

ˇr21

!

Vp1qj ă r2

)

´ r11

!

Vp1qj ă r1

)ˇ

ˇ

ˇ

ď |r2 ´ r1| ` 1

!

Vp1qj ă r2

)

|r2 ´ r1| ` r1

ˇ

ˇ

ˇ1

!

Vp1qj ă r2

)

´ 1

!

Vp1qj ă r1

)ˇ

ˇ

ˇ

Taking the expectation, we obtain

Eˇ

ˇπpξ j, r1q ´ πpξ j, r2qˇ

ˇ ď |r2 ´ r1| ` |r2 ´ r1|PrpVp1qj ă r2q ` r1Prpr2 ă Vp1qj ă r1q


“ Op|r2 ´ r1|q ` f1pr2qpr1 ´ r2q ` opr1 ´ r2q “ Op|r2 ´ r1|q.

This establishes (3).

Finally, we derive Σ. Notice that:

Erπp¨, rqs “ Erπp¨, rqs Érπp¨, r˚qs

“

ż ω

0maxtr, Vp2qu f2puqdu´

ż ω

0r1tVp1q ă ru f1pvqdvÉrπp¨, r˚qs

“ rż r

0f2puqdu`

ż ω

ru f2puqdu´ r

ż r

0f1pvqdvÉrπp¨, r˚qs

Differentiating this expression with respect to r, we obtain

BErπp¨, rqsBr

“

ż r

0f2puqdu` r f2prq ´ r f2prq ´

ż r

0f1pvqdv´ r f1prq

“

ż r

0f2puqdu´

ż r

0f1pvqdv´ r f1prq

Taking the second derivative and evaluating at r˚, we obtain

Σ “ ´ f2pr˚q ` 2 f1pr˚q ` r˚ f 11pr˚q.

�

Proof of Theorem 3

Proof. This theorem follows directly from Theorem 2. First, notice that pprJq ´ ppr˚q “1JřJ

j“1 πpξ j, rJq ` ppr˚q ´ ppr˚q. Since rJ ´ r˚ “ OPpJ´1{3q and 1JřJ

j“1 πp¨, r˚ ` αJ´1{3q “

OPpJ´2{3q, the term 1JřJ

j“1 πpξ j, rJq is also OPpJ´2{3q. A simple application of the Central

Limit Theorem establishes the result. �

Proof of Theorem 4 We will use slightly different notation in this proof than elsewhere

in the paper, letting SJ “ pz1, . . . , zJq be a fixed sample of size J and denoting quantities

estimated on this sample by a subscript SJ .

We start with the definition of empirical Rademacher complexity:

Definition. Let G be a family of functions from Z to ra, bs, and SJ “ pz1, . . . , zJq a fixed sample

of size J with elements in Z. Then the empirical Rademacher complexity of G with respect to SJ is


defined as:

pRSJ pGq “ Eσ

»

–supgPG

1J

Jÿ

j“1

σjgpzjq

fi

fl

where σ “ pσ1, . . . , σJq, and each σj is an independent uniform random variable with values in

t´1,`1u.

We state a useful preliminary result.

Theorem 5. Let G be a family of functions mapping Z to r0, 1s. Then for any δ ą 0, with

probability at least 1´ δ, for all g P G:

Ergpzqs ´1J

Jÿ

j“1

gpzjq ď 2pRSJ pGq ` 3

d

log 2δ

2J

Proof. See Mohri et al. (2012), Theorem 3.1. �

This result can be straightforwardly adapted to obtain a two-sided bound.

Corollary 1. Let G be a family of functions mapping Z to r0, 1s. Then for any δ ą 0, with

probability at least 1´ δ, for all g P G:ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Ergpzqs ´1J

Jÿ

j“1

gpzjq

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ď 2pRSJ pGq ` 3

d

log 4δ

2J

Proof. Applying Theorem 5 above to G1 “ t´g ` 1 : g P Gu and noting that pRSJ pGq “pRSJ pG

1q, we obtain the result that for any δ{2 ą 0, with probability at least 1´ δ{2, for all

g P G:

1J

Jÿ

j“1

gpzjq ´ Ergpzqs ď 2pRSJ pGq ` 3

d

log 4δ

2J.

Theorem 5 also implies that for any δ{2 ą 0, with probability at least 1´ δ{2, for all g P G:

Ergpzqs ´1J

Jÿ

j“1

gpzjq ď 2pRSJ pGq ` 3

d

log 4δ

2J.

Combining these two results and applying the union bound gives the desired result. �


Now for simplicity let v0 “ 0 so that πpVp1qj , Vp2qj , rq “ r1pVp2qj ă r ď Vp1qj q `Vp2qj 1pr ď

Vp2qj q and G ” tπp¨, ¨, rq : r P r0, ωsu. We can now prove an upper bound on ppr˚q ´ ppprSJ q

in terms of the empirical Rademacher complexity of G.

Lemma 1. Let 0 ď Vp1qj ď ω ă 8. For any δ ą 0, with probability at least 1´ δ, it holds that

ppr˚q ´ ppprSJ q ď 4pRSJ pGq ` 6ω

c

log 4δ

2J .

Proof.

ppr˚q ´ ppprSJ q “ ppr˚q ´ ppSJ pprSJ q ` ppSJ pprSJ q ´ ppprSJ q

ď ppr˚q ´ ppSJ pr˚q ` ppSJ pprSJ q ´ ppprSJ q.

ď 2 suprPR

|pprq ´ ppSJ prq|.

The first inequality follows because prSJ maximizes ppSJ by definition. Applying Corollary 1

with zj “ pVp1qj , Vp2qj qwe have that for any δ ą 0, with probability at least 1´ δ:

suprPr0,ωs

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1ω

ErπpVp1qj , Vp2qj , rqs ´1

Jω

Jÿ

j“1

πpVp1qj , Vp2qj , rq

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ď2ωpRSJ pGq ` 3

d

log 4δ

2J,

or equivalently,

suprPR

|pprq ´ ppSJ prq| ď 2pRSJ pGq ` 3ω

d

log 4δ

2J.

Therefore for any δ ą 0, with probability at least 1´ δ:

ppr˚q ´ ppprSJ q ď 4pRSJ pGq ` 6ω

d

log 4δ

2J.

�

Following Mohri and Medina (2016), define π1pVp1qj , Vp2qj , rq “ Vp2qj 1pr ď Vp2qj q` r1pVp2qj ă

r ď Vp1qj q `Vp1qj 1pVp1qj ă rq and π2pVp1qj , rq “ ´Vp1qj 1pVp1qj ă rq, so that πpVp1qj , Vp2qj , rq “

π1pVp1qj , Vp2qj , rq ` π2pV

p1qj , rq. Define also G1 “ tπ1p¨, ¨, rq : r P r0, ωsu and G2 “ tπ2p¨, rq :

r P r0, ωsu. The following lemma is useful:


Lemma 2. Let H be a set of functions mapping X to R and let Ψ1, . . . , ΨJ be µ-Lipschitz functions

for some µ ą 0. Then for any sample SJ of J points x1, . . . , xJ P X , the following inequality holds:

1J

Eσ

»

–suphPH

Jÿ

j“1

σjpΨj ˝ hqpxjq

fi

fl ďµ

JEσ

»

–suphPH

Jÿ

j“1

σjhpxjq

fi

fl .

Proof. See Lemma 14 in Mohri and Medina (2016). �

We now find an upper bound for the right hand side of Lemma 1, which is not expressed

in terms of Rademacher complexity and which makes the asymptotic behavior of the term

ppr˚q ´ ppprSJ q clear. This will lead to Theorem 4.

Lemma 3. Let 0 ď Vp1qi ď ω ă 8. Then pRSJ pGq ď2ω?

log 2J `ω

b

2`2 log JJ .

Proof. Note that pRSJ pGq ď pRSJ pG1q ` pRSJ pG2q, as the supremum of a sum is less than the

sum of suprema. We give upper bounds on both of these terms. For the first term, we

have:

pRSJ pG1q ” Eσ

»

– suprPr0,ωs

1J

Jÿ

j“1

σjrVp2qj 1pr ď Vp2qj q ` r1pVp2qj ă r ď Vp1qj q `Vp1qj 1pVp1qj ă rqs

fi

fl

ď1J

Eσ

»

–suprPR

Jÿ

j“1

σjr

fi

fl

“1J

Eσ

»

– suprPt0,ωu

Jÿ

j“1

σir

fi

fl

ď2ω

a

log 2J

The first inequality follows from applying Lemma 2 with Ψjpxq ” π1pVp1qj , Vp2qj , xq and

hpxq ” x, and the observation that the functions Ψjprq are 1´Lipschitz for all j. The equality

follows because the supremum will always be attained at r “ 0 (ifřJ

j“1 σi ď 0) or at r “ ω

(ifřJ

j“1 σj ą 0). The final inequality is an application of Massart’s lemma (see, for example,

Mohri et al. 2012).

For the second term, we have

pRSJ pG2q ” Eσ

»

– suprPr0,ωs

1J

Jÿ

j“1

´σjVp1qj 1pVp1qj ă rq

fi

fl


ďω

JEσ

»

– suprPr0,ωs

Jÿ

j“1

´σj1pVp1qj ă rq

fi

fl

“ω

JEσ

»

– suprPr0,ωs

Jÿ

j“1

σj1pVp1qj ă rq

fi

fl

ď ω

d

2` 2 log JJ

.

The first inequality follows from applying Lemma 2 with Ψjpxq ” Vp1qj x and hpxq ”

1pVp1qj ă xq, noting that Ψjpxq are ω-Lipschitz for all j. The equality follows because the

distributions of σj and ´σj are identical. Finally, the last inequality follows from Massart’s

lemma (see Proposition 2 in Mohri and Medina 2016).

Putting the bounds on pRSJ pG1q and pRSJ pG2q together, we have:

pRSJ pGq ď2ω

a

log 2J

`ω

d

2` 2 log JJ

.

�

This leads immediately to Theorem 4:

Theorem. Let 0 ď Vp1qj ď ω ă 8. For any δ ą 0, with probability at least 1´ δ over the possible

realizations of SJ , it holds that

ppr˚q´ppprSJ q

ω ď

˜

8?

log 2J ` 4

b

2`2 log JJ ` 6

c

log 4δ

2J

¸

.

Proof. Combine Lemmas 1 and 3. �

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